My ds4 works out of the box— the touchpad even maps to the mouse cursor (I use libinput and Wayland).
Ds4drv, if you use it, will override the default kernel driver I believe. Make sure it’s not emulating an Xbox 360 controller, unless you want that.
ArchWiki also mentions that you need to disable hidraw in Wine. Maybe that’s related, if you have issues with Windows games?
I’m sorry to hear you’re having trouble!
Regarding the vainfo thing, at least on Arch, the Nvidia vaapi driver is a separate package. I’m not sure what the equivalent is on bazzite, but that’s likely why vainfo doesn’t show anything. And you should still be able to use nvenc anyways, which sunshine will choose before vaapi.
Is moonlight able to see the computer on your network? And in sunshine’s web interface under the “troubleshoot” tab, there’s a log. Does it say anything that might help?